Recursos de colección

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Székelyhidi, Gábor; Tosatti, Valentino; Weinkove, Ben
We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern–Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.

Mikhalkin, Grigory
We associate a half-integer number, called the quantum index, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to $\pi^2$ times the quantum index of the curve, and thus has a discrete spectrum of values. We use the quantum index to refine enumeration of real rational curves in a way consistent with the Block–Göttsche invariants from tropical enumerative geometry.

Kleiner, Bruce; Lott, John
We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. These provide a solution to the long-standing problem of finding a good notion of Ricci flow through singularities, in the $3$-dimensional case. ¶ We prove that Ricci flow with surgery, starting from a fixed initial condition, subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows.

Kalmykov, Sergei; Nagy, Béla; Totik, Vilmos
Asymptotically sharp Bernstein- and Markov-type inequalities are established for rational functions on $C^2$ smooth Jordan curves and arcs. The results are formulated in terms of the normal derivatives of certain Green’s functions with poles at the poles of the rational functions in question. As a special case (when all the poles are at infinity) the corresponding results for polynomials are recaptured.

Brendle, Simon; Choi, Kyeongsu; Daskalopoulos, Panagiota
We consider a $1$-parameter family of strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ moving with speed $-K^{\alpha} ν$, where ν denotes the outward-pointing unit normal vector and $\alpha \geqslant 1 / (n+2)$. For $\alpha \gt 1 / (n+2)$, we show that the flow converges to a round sphere after rescaling. In the affine invariant case $\alpha = 1 / (n+2)$, our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.

Knop, Friedrich; Krötz, Bernhard; Schlichtkrull, Henrik
Let $G/H$ be a unimodular real spherical space which is either absolutely spherical, i.e. the real form of a complex spherical space, or of wave-front type. It is shown that every tempered representation for $G/H$ embeds into a twisted discrete series for a boundary degeneration of $G/H$. If $G/H$ is of wave-front type it follows that the tempered representation is parabolically induced by a twisted discrete series representation for a real spherical space formed by a Levi subgroup.

Huh, June; Wang, Botong
One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erdős: Every set of points $E$ in a projective plane determines at least $\lvert E \rvert$ lines, unless all the points are contained in a line. The result was extended to higher dimensions by Motzkin and others, who showed that every set of points $E$ in a projective space determines at least $\lvert E \rvert$ hyperplanes, unless all the points are contained in a hyperplane. Let $E$ be a spanning subset of an $r$-dimensional vector space. We show that, in the partially ordered...

Bonahon, Francis; Dreyer, Guillaume
For a closed surface $S$, the Hitchin component $\mathrm{Hit}_n (S)$ is a preferred component of the character variety consisting of group homomorphisms from the fundamental group $\pi_1(S)$ to the Lie group $\mathrm{PSL}_n (\mathbb{R})$. We construct a parametrization of the Hitchin component that is well-adapted to a geodesic lamination $\lambda$ on the surface. This is a natural extension of Thurston’s parametrization of the Teichmüller space $\mathcal{T}(S)$ by shearing coordinates associated with $\lambda$, corresponding to the case $n=2$. However, significantly new ideas are needed in this higher-dimensional case. The article concludes with a few applications.

Laza, Radu; Saccà, Giulia; Voisin, Claire
Let $X$ be a general cubic $4$-fold. It was observed by Donagi and Markman that the relative intermediate Jacobian fibration $\mathcal{J}_U/U$ (with $U=(\mathbb{P}^5)^\vee\setminus X^\vee$) associated with the family of smooth hyperplane sections of $X$ carries a natural holomorphic symplectic form making the fibration Lagrangian. In this paper, we obtain a smooth projective compactification $\overline{\mathcal{J}}$ of $\mathcal{J}_U$ with the property that the holomorphic symplectic form on $\mathcal{J}_U$ extends to a holomorphic symplectic form on $\overline{\mathcal{J}}$. In particular, $\overline{\mathcal{J}}$ is a $10$-dimensional compact hyper-Kähler manifold, which we show to be deformation equivalent to the exceptional example of O'Grady. This proves a conjecture...

Fang, Fuquan; Grove, Karsten; Thorbergsson, Gudlaugur
There is a well-known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns–Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least $3$ and irreducible topological spherical buildings of rank at least $3$. ¶ We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type $\mathsf{M}$ associated with any polar action of cohomogeneity at least $2$ on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type $\mathsf{M}$, its...